246 PROCEEDINGS OP THE AMERICAN ACADEMY 



the totality of equations (1) to (12), we obtain the functional equations 



(17) Fti/fra), /*)=/,(*, *0*,o)) 



(» = 1, 2 



We have already denoted by T„ the transformation defined by (1), in 

 which the parameters arc <i^ a... Consequently, equations <i.'n define 

 the transformation /',. We may now denote the transformation defined 



by (10) in which the parameter, are /<>• /'■, l'.\ /'.. Then the functional 

 equations (17) may he expressed in the single formula 



(18) T a E.= r„. 



For T & transforms x t into j;'=f(x, a), and E^ transforms x i into 

 ^i ( x '> /*)» while, in virtue of (12), T„ can also he written in the form 



x t '=f t (x, <f>(/x, a)) (t=l, 2). 



The relation (18) persists, therefore, provided the three parameter 

 systems (a u a,), (^ ^ 2 ), (a v a 2 ) are connected by relations (12). 

 Therein a„ a 2 denote definitely chosen general values of r/„ Oj. 



We now make use of the assumption that the transformation 7',, de- 

 fined hy (1), shall hecome the identical transformation for a y — a 

 « 2 = a 2 (0) . Namely, for the moment, let a x = a^ *, u 2 = a 2 " ; whereupon 

 T r , becomes the identical transformation 7^(0). Then, because the deter- 

 minant of the a jk (a' ') is, according to assumption, neither zero nor in- 

 finite, and, therefore, the former considerations are also valid fur a = a !0) , 

 — the a v a., in virtue of (12) assume the values 



(19) a k = * k (n, to, <> a.n (k = 1, 2) ; 



that is to say, since we take a x (0) = er 2 ' n ' = 0, a x and a 2 assume the values 



Whence it follows that each transformation E^ belongs to the family of 

 transformations T,,, defined b;/ equations \\). If now, conversely, we 

 could establish that each transformation 7], belonged to the family of 



